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Gregg showed that if we (a) treat taxa as sets defined by extension (i.e., by listing all members), and (b) accept that two sets with exactly the same content must be the same set, then many biological classifications violate these premises because the same taxon may be assigned to multiple levels in the Linnean hierarchy. For example, the aardvark, Orycteropus afer, is the only extant species of the genus Orycteropus, which is the only extant member of the family Orycteropodidae, which in turn is the sole extant representative of the order Tubulidentata. Under Gregg's model, Tubulidentata, Orycteropodidae, and Orycteropus are all the same thing as they have exactly the same content (i.e., Orycteropus afer). Put another way, monotypic taxa are redundant and violate basic set theory. Gregg would argue that they should be eliminated.

Wikipedia illustrates this nicely. Wikipedia conforms to Gregg's model in that taxa are defined by extension (each taxon comprises one or more wiki pages), and if taxa have the same content only one taxon (typically that with the lowest taxonomic rank) has a page in Wikipedia. Put another way, if the aardvark is the sole representative of the Tubulidentata, then there is nothing that could be put on the Tubulidentata page that shouldn't also belong on the page for the aardvark. As a result, the page for the aardvark gives a full classification of this animal, but most taxa in the hierarchy don't have their own pages.

Responses

There are several possible responses to Gregg's paradox. One is to argue that taxa should be defined intensionally (i.e., on the basis of their characters), which was Buck and Hull's approach. Essentially, they were arguing that we could (somewhat arbitrarily) specify properties of Orycteropodidae that weren't shared by all Tubulidentata, and hence we are justified in keeping these taxa separate. Gregg himself was less than impressed by this argument (doi:10.2307/2412017).

Another approach is to suggest that we may discover taxa in the future that will, say, be members of Orycteropus but which aren't O. afer, and that the taxa between the rank suborder and species are placeholders for these discoveries. Indeed, in the case of the Tubulidentata there are extinct aardvarks (doi:10.1163/002829675x00137, doi:10.1016/j.crpv.2005.12.016, and doi:10.1111/j.1096-3642.2008.00460.x) that could be added to Wikipedia, thus justifying the creation of pages for the taxa that Gregg would have us eliminate.

Of course, Gregg's paradox is a consequence of having ranks and requiring each rank (or at least a reasonable subset of them) to exist in a classification. If we ignore ranks, then there's no reason to put any taxa between Afrotheria and Orycteropus afer. So, we could drop this requirement for having taxa at each rank or, of course, drop ranks altogether, which is one of the motivations behind phylogenetic classifications (e.g., the phylocode).

Implications for parsing Wikipedia

From a practical point of view, Gregg's paradox means that one has to be careful parsing Wikipedia Taxoboxes. As I've argued earlier, the simplest way to ensure that a classification is a tree is for each taxon to include a unique parent taxon. The simplest way to extract this for a taxon in a Wikipedia page would be to retrieve the taxon immediately above it in the classification (i.e., for Orycteropus afer this would be Orycteropus). But Orycteropus doesn't have a page in Wikipedia (OK, it does, but it's a redirect to the page for the aardvark). So, we have to go up the classification until we hit Afrotheria before we get a taxon page.

Personally I quite like the fact that a largely forgotten argument from the middle of the last century concerning logic and Linnean taxonomy seems relevant again.